Question

given: ∠kjl and ∠kml intercept arc kl. prove: m∠kjl = m∠kml

statements | reasons
--- | ---

1. ∠kjl intercepts \\(\widehat{kl}\\) | 1. given
2. ∠kml intercepts \\(\widehat{kl}\\) | 2. given
3. \\(m∠kjl = \frac{1}{2} (m\widehat{kl})\\) | 3. ♣
4. \\(m∠kml = \frac{1}{2} (m\widehat{kl})\\) | 4. ♦
5. \\(m∠kjl = m∠kml\\) | 5. ♠

(image of a circle with points j, m, l, k on it)

♣ = dropdown with options: angle formed by a tangent and a chord thm., inscribed angle thm., second corollary to the inscribed angle thm., substitution property
♦ =
♠ =

Explanation:

Step1: Reason for Statement 3

The inscribed angle theorem states that an inscribed angle is half the measure of its intercepted arc. Since \( \angle KJL \) is an inscribed angle intercepting arc \( KL \), we use the inscribed angle theorem. So the reason for \( m\angle KJL=\frac{1}{2}(m\widehat{KL}) \) is the inscribed angle thm.

Step2: Reason for Statement 4

Similarly, \( \angle KML \) is also an inscribed angle intercepting arc \( KL \). By the same inscribed angle theorem, \( m\angle KML = \frac{1}{2}(m\widehat{KL}) \). So the reason here is also the inscribed angle thm.

Step3: Reason for Statement 5

We know from statements 3 and 4 that \( m\angle KJL=\frac{1}{2}(m\widehat{KL}) \) and \( m\angle KML=\frac{1}{2}(m\widehat{KL}) \). By the substitution property (since both are equal to \( \frac{1}{2}(m\widehat{KL}) \), we can substitute one for the other), we get \( m\angle KJL = m\angle KML \). So the reason is substitution property.

Answer:

• Reason for Statement 3: inscribed angle thm.
• Reason for Statement 4: inscribed angle thm.
• Reason for Statement 5: substitution property.